Method and apparatus for solving mathematical problems



F. J. MURRAY July 18, 195o f METHOD ANDAPPARATUS-FOR SOLVINGMATHEMATICAL PROBLEMS 5 sheets-sheet 1 'Filed Apri1 15, 1948 QrllIIINVENTOR W. e. Uhm/5 Mvm Mm lvm C MA ...f

F. J. MURRAY July 1s, 195o METHOD AND 'APPARATUS FOR SOLVINGMATHEMATICAL PROYBLEMS Filed April 15,` 1948 ATTORNEYS Jlr 18, 195,0 F.J. MURRAY v v METHOD AND APPARATUS FOR SOLVING MATHEMATICAL PROBLEMSFiied April 15, 1948' 5 Sheets-Sheet 3 www BY da ATTORNEYS July 18, 1950F. J. MURRAY METHOD AND APPARATUS FOR SOLVING MATHEMATICAL PROBLEMSFiled April 15, 1948 55K /aa 5 Sheets-Sheet 5 HMME TE R INVENTORATTORNEYS Patented July 18, r195() T .ori-ice METHOD AND APPARATUS ORSOLVING MATHEMATICALPROBLEMS Francis J. Murray, New York, N. Y.

Application April 15, 1948, Serial N0. 21,215 r (C1, :m5- 61) v 20Claims. l

The present invention pertains to vimprove- 4 ments in methods andapparatus for solving mathematical problems.

An object of the invention is to provide a method of solvingsimultaneous linear equations by combining arbitrary quantitativephysical factors representative of the various unknowns and theircoeflicients in such a manner as to produce a single indicationrepresentative of a combined initial error factor, and varying thequantities representing the unknowns until the indicated error becomeszero, whereby the nal entered quantities representative of the unknownsmay indicate the solution of the problem.

Another object is to lprovide suitable apparatus for carrying out themethod.

A further object is to provide apparatus of the above nature operable byelectricity.

A still further object is to provide suitable electrical apparatus ofthe above type in two categories, ynamely a first. embodiment which isof maximum simplicity and cheapness and is `adapted to yield rresultswith generally practical or slide-rule accuracy, and an alternativeembodiment in which accuracy may be carried to the highest degree ofprecision.

Other objects and advantages will become evident during the course ofthe following description in connection with the accompanying drawings,in which:

Figure 1 is a front elevation of an instrument embodying the simplifiedform of the invention; Figure 2 is a schematic or iiow vdiagram showingthe general arrangement lof the functional sub-assemblies; v

Figure 3 is a detail, schematic wiring diagram of the type of instrumentshown in Figure 1;

Figures 4 and 4a. show a similar schematic Wiring diagram' of thealternative or high-precision type of instrument; and

Figure 5 is a detail diagrammaticillustration of the substitution ofcapacitative conductances for resistive conductances in the coefficientgroups, Figure 4.

In order to promote clarity in explaining the purpose and operation ofthe apparatus itself, the nature and properties of the mathematicalequations to be-solved will rst be set forth. ,Y 'For maxe;

imumv simplicity the demonstration apparatus herein described is devisedfor the solution of two simultaneous equations involving two variables Xand Y. It Will become evident however, that the invention is applicableto any desired number of equations and .variab1es`. y

in which y y X and Y are the required unknowns;

am and aai are,V the coefficients of X in the first and second equationsrespectively;v

:11,2 and aaa arethe coefcients of Y in the two equations; and

b1 and b2 are the constants in the two equations.

Since the solution must be carried out with instruments of iinite rangeand since Widely varying magnitudes of 'unknowns are to be encounfteredin solving different problems, it is desirable to introduce a third orgage variable T, by which the scale of the solving process may be keptwithin the practical scope of the instruments and by which the mostprecise utilization of the latter may be obtained. f

Introducing the gage variable 'T converts the above equations to If bothe1 and e2 can be reduced to zero, the last written equations may berewritten Comparing these forms withthe original equation, it will beevident that withzero condition of e1 and e2,

tex

and

that is and constitute thesolution of the original problem. l

, The factors e1 and E2 give a Working medium for;4 solving the problem,since they canbe i realized A calibrated to permit entry of` the desiredvalue of 411,1. The sign of the entered coefficient may be made eitherplus or minus by means of the reversing switch 65 which interchanges theconnectionsof group B3 with` the leads 41 and 48.

From the foregoing it will be seen that thepoground The micro-ammeter`II` is `connected across the two cathodes H9 and |2001 the tube tentialof the output lead 'I2 represents the mathl ematical expression:

amm cos wt 1 Similarly, the output potentiometer I5 of group 'I3 has acalibrated setting dial 8|, Figure l, and

v 'the input. fSince the two inputs to the diode I I0 the potential ofthe output lead 16 represents the expression amm cos wt.

The numerals 182,203, 84 and v85, Figure 3, represent`four moreresistance groups identical in structure and having outputpotentiometers 816, 81, 80, 89 and reversing or sign switches 90, 9|,92, 93.

The groups 82 and i83 are connected in parallel to the output conductors59 and 60 of the 'y vlari= able entry group 34, and by the same processdescribed with respect to groups 63 and i64, the potentials of theoutput leads 94 and 95 of groups 02 and B3 are respectively amy cos wtand aaai/ cos wt, where y is represented by the setting of the inputpotentiometer 3| )and 111,2 and am by the settings of outputpotentiometers 486` and `181.

Similarly, the groups E84 :and 815 are connected in parallel to theoutput leads 6I and 02 of the T or gage variable entry group 35, and thepoten` tials of their output `leads 96 and 91 respectively are lnT coswt and bzT cos wt, b1 and b2 being represented by the settings ofVoutput potentiometers 081and89. l

The output leads 12, 94 and 96 are connected to a common conductor 98through parallel matched resistors each comprising a xed wirewoundresistor 99 and a variable-resistor I00 provided for fine .adjustmentinmjatching. By this means the three potentials are averaged, yielding avoltage in conductor 9B proportioned to (dni-l-qrzy-ljbiT) COS wir-rercos wt and since cos wt is a factor :common to both sides,

the rvoltage of conductor 98 is proportional to t potentialsof outputleads 16, 95 and 9'!` are avere: aged through three parallel matchedresistor' combinations I0l connected to a common conductor I02 having apotential proportional to The conductors 98 and I 02 Iare connectedrespectively to` the grids I`03 and |04 of a double triode tube I05having corresponding `plates |05 and I0'I. Thetube I05serves as anamplifier, and

as the various connectionsof such tubes lare well.l

sistor -I IIl, the resistance :of II4 being equal to the effectiveparallel resistance of resistors III and II2. The potential of thecommon conductor is impressed via a lead I I5 on onegrid `I I'B of `asecond double` triode` amplifier tube II'I. The 'ode'.Il'I is connectedto` second grid I |13 o Y outpurcurrjent of the diode un are aredirectly proportional via the double triode I 05 1 to theinputpotentials E1 and ez of the latter, it

follows that the outputs of diode IIO are proportional to 612 and e22.Furthermore, the potential of the common output conductor I |13, whichcarries the sum of the two outputs of diode IIO is proportional tothesum of G12 and E22 and since as noted, changes in the reading ofmicro-ammeter jI I are directly proportional to changes in the potentialof conductor H3, the changes in the reading give an indication of thechanges in 6124-622.

e Since the `object throughout is not to determine electrical`quantities but rel-ations between them, `as entered in terms of thevariables :c and y, the reading of the meter II may be interpreted asrepresenting the term ,u in the expression Again, since the solution ofthe problem involves the elimination of the error factors e1 and e2, thesolution depends not on a specic positive value reading `of a, but onits reduction to zero. Iihus the use ofthe amplier tubes I05 and II1does not change the proportions of the variables a: and y or the errorfactors e1 and f2, but simply serves to increase the amplitude of themovement of the meter II, whereby the minimizing and reduction of a tozero may be observed with maxiu mum precision.

The relationship of the vario-us 1 described groups of apparatus andtheir functions in the derivation of a are illustrated in simplifiedform on the ow diagram, Figure 2. Since the two equational circuitsremain functionally separate until their combination in the nal additionnetwork l I I, II2, II3 and IM, the double triode tube |05 and thedouble diode IIO are shown on the olwdiagra-mby two separate boxes each.Actually a pair each of single triodes and single diodes may be used ifdesired, but the double tubes are preferred in practice for maximumsimplicity.

Operation The generalmethod of operation is to enter trial values of theunknowns a: and y, enter a tentative value of the gage variable T, enterthe known values of the coefiicients 511.1, are, am, bi and Ab2 with theproper signs, observe the indica-- tion of /i on the meter II, then varythe settings -of m and yuntil the indicated a becomes zero. If

upon initial entry it appears that the entered lvalue of, T is not suchas to place the operation Vin proper range of the instruments, adifferent ,e the settings of n: and y are divided by the setting of T,and the two quotients give the desired solutionlof the originalEquations l and 2, since as previously pointed out, when e1 and e2 each0,

y ,It isclear that any method ofadjusting `:1: and y vhlshe'svlts' in,can be ,used in this. device.

`pass through resistors 22 and 23 in series.

weg-sirenas vvOne such method is th-following'. One"-entry, Vfor-exampley, is'left constant and y:a: is-varied until the indicated p. isminimized, that is until 1 4any further variation of .t in any directioncauses anincrease in fr. Then a: is heldA constant and y is varied-untilc is -minimized'relative to y. This process is continued, alternatingbetweenand y until p is not sensibly diierentvfromtzero.

` `The strength of the electrical-signall/ris determined by the controlnetwork 2I. Initially a onthe meter. To compensate for this, thejsignalstrength is increased by shuntingthe various resistors in the network 2I.

`of the device to .,u and consequently toftheerror .factors ei'and e2 isincreased as'fthese'become Smaller.

Thus the sensitivity Simply stated, with respecttothe'twoequationsbeingused as examples, the'v above; method makes useof the fact-that when incorrect ltrial entries of the unknowns :c and yresult inerrors 'el and en in the two simultaneous equations, the

nearer either variable may be brought to its correctvalue, the less willbecomethe errors' and consequently the sum of their squares asindivcated by ,u=e12|e22 While for the purposes of simplicity. thefore-`going explanation .was .carried out withrespect to the two equationsof. two variables and related apparatus under immediate illustrativeconsider- I ation, it-will be proved mathematically hereinafter that themethod of successive minimization of ,l withrespect to unknownsmaybecorrectly a -'dial'= position faridl the-inputvswitch I 4a, isfclOSed,

energizing the determining apparatus.

A tentative setting f T is `made on'the dial ofipotentiometer 32,-forexample .2.' Thefinput control switches* being in low input` position,

@only La-sr`nall1va1ue of ,uappears on the meter II.

""fIhe-'inputacontrol isshifted to medium posiweak signal is sufficientto indicate #but as vthe error factors e1 and e2 are -lessened intheadjusting process, p assumes smaller and smaller values tion and p thenreads approximately 15v (the exact numerical-.value isnot significant)which is a large enough dial ligure to work With properly.

" LThe :v setting is then varied, to minimize fr,

Mand resultsin'u the" readings :1:=.25, p25. The

y setting listhen varied to-give a rough balance, and'ity is Afound thatwhen y=.1, a is reduced to 1, which indicates that the range isunnecessarily low; that is,- T is too small. To vary Tand preservetheroughY balance, it is necessary to multiply the values of fr, y and T bythe same factor, andl as the rough balance indicates the generalproportions between as, y 'and T', it is necessary that the` multiplyingfactor be small enough'not to throw; the' values of fr or vy off scale.For example, if Twere taken as 1., that isve times its` linitial value,:I: would have to be raised from .25 to1.25,which would be oli'l itsscale which q extendsl only tounity in the illustratedv device. 2Therefore T lis? taken as .5, which calls for settings `of a,`v and y at`.62:and .'25 respectively, to

maintain the; approximate balance; With these settings fr appears as 4.

'cc is now 'adjusted' to minimize ,getting .r=.6,

ythen 'yA is adjusted to minimize ,u at which applied to any desirednumber of variables and corresponding s-imultaneous linear equations.

The successive actual steps taken vrinfsolving equationsas outlinedabove may best lloe..illus ytrateclby an .example as follows:

' .The input switch Illar is first -opened and-.the switches24 and 25`are also placedopenposition.. This position of bothswitches 24.and. 25will hereinafter be referred to as low input position, as it requiresvany circuit current to A secondposition, in which-switch 2li is @closed.and switch 25 opened will be termed medium input positiomsince. onlyresistor 23 mustrbe traversed by the current.y Closedzpositionl of bothswitches 2liv and 25 will be .termed-.Thigh input position, as itley-passes..both-resistors and allows-full line voltage to be -appliedto :the

determining apparatus.

IWith input switch Ma, open, theline switch I4 is` closed, energizing-the power supply-and rectifying unit I8 to warm up the varioustubes oof theapparatus.y During this-.process.the coeflicient potentiometersS1, l5, 86,.81, .88 and 89 are shifted respectively to .dial `positionsyof .4, .5, .8, .7, .8 and .32 and the reversing orsign switches 9|, 92and 93 are thrown to minus" position, since .inthe given..V equationsthe c0- eiicients b1(=;.8)., a2,2(="`-".'7), and b2(=.32)

Acarry the negative sign.

'The x potentiometer 30, theypotentiometer -3I vand the T potentiometer32 are set tozero point y=.2.-'The meter reading p is now apparentlyzero, indicating that the correct :c and y settings havev `beenachieved. "However, as a nal test ofl the balance, 'therv input controlswitches may be Lthrown to fhighl input position. 'Under these:conditions any small residual errors inthe :c and/or y settings'cause alarge increase inw, for example, to'an indicationof 12.

f the 'minimum-reading which can be secured on thev sol meter I I- is22instead of absolute scale zero.V This constant deviationfrom scalezero vrepresents the highly magnied indication of the small lossesinherent in any thermionic system, and may be .termed the slideruletolerance of the system. It is so innitesimal as vto beindistinguishable in normal or lmedium` inputWOperating range and in thehigh input range has only theeii'ect -j fof-fv shifting'theoperationalzero' point of lrfrom scaleiizero to-scale 2 in the nal or Vernieradiustment of rand y.

'For'. further conveniencey in 'illustrating theabove'fdescribedprocedure, the steps are listed in tabular form' asAlfollows:

' setT '2....

ry .Raise Input Balance.

Step Input Strength The apparatus and` method'have been" illustrativelyset forth as applied to'two equations ci?v two variables, but the devicemay be constructed for any desired number of variables and correspondingequations, the method orminimizing of u with respect to successivevariables always providingthe required solution. This fact may bedemonstrated mathematically Aas follows: i v

noted,

becomes vless'than a prescribed quantity points in a single cycle.

Now at Pft),

`for allj For input e, the output f(e) of a diode tubehas thecharacteristics of e2, that is (a.) It is an even function of e whi whenand only when :0.

(b) f (the first derivative) exists at every A value of e and is zero at5:0 and only there.

(c) f" (the second derivative) exists and y is bounded from below inevery nite interval by a constant bound greater than zero.

Under these hypotheses, consider n mi az, corresponding to n equationsThis indicates that there is some interval in` which all the em) and 0(zero) are contained. Let C denote the constant 0 which is the lowerbound for f" on this interval.v

For a point {1:1 intervaL.

Using Taylors theorem,

Now, at point PM) in consideration,= y

v si and the remainder derived rem becomes Thus since the values of uw)converge, for values of the S eld sufficiently large the points PW) PGMina single cycle may be taken closer together than any prescribedamount.

In afinite region is uniformly eontinueus-g-spreviously stated-andch iszero i variables :cnafbe the point in the's: l cycle of successive.minimizations `which results will, since chas been* i o 45 Referring toFigure 4, the structure where K isl a constant 0 which depends only uponthe given set of coeicients. o l l i from Taylors theo- If, as shown,these values are very small, and since the am matrix is not singular,all values Qf f (45M) must be small, which in turn indicates i "that allvalues of JM) are small. Thus at point variable xt, the error factorbecomes smaller than i mization of ,i with respect to the othervariables,- the errors are eliminated one `by one to arrive at rlthecomplete solution. i The embodiment of the invention'illustrated inmanner already described, contains certain modifications in the meansarriving at the values` of l o In this embodiment the coefllcient re-r';

sistance groups have been arranged to allow the coeicients to be entereddigitally by closing-f` switches instead of by adjusting potentiometers,thus `permitting the entries to be made by any; desired switchingcontrol means, leither manual or by punched cards and the like.

As in the `case of Figure, the :12, y and T vari," able groups |2I, |22and |23 are of similar con-v struction, so that description of the xgroup |2| n may be taken as typical of all three. In the same manner thea1,1 coefcientgroup |24, the

, identically constructed, and am group |24 will be `described astypical of all.

for the reversing switch 46 have connected across vthem in series a setof fourequal resistors |32,

"5031.33, |34 and las. A center tap las 1eadsi0 ground at |31, Whileintermediate taps |38 and,i

'7 s ff'or potentials ,-l-m and -v in the leads |30 `and; |3I, thepotentialsof leads |38 and |36` and l'39 will be respectively .5:r, 0and -.5a:.

In vthe am coeiicient group |24, the lea switch |40 of anyosuitabletypeito a xed resistor |4I, thence to an si output conductor |42.

Five additional live-point selector switches |43,-

|44, |45, |46 and |41 are similarly adapted to mon output conductor |42.[48,1149, |50, (5| and |52 havedifferent xed valuesincreasing'progres'sively as illustrated for clarity by value guresvo'nthediagram.

By the above arrangement a'large number of A setresistances in gradedcombinations can bei," connected as coeicient'ari between the .r outputj leads" |30 and "13| andthe ei-output conductor' at point Ps it iszero. Thuswith siarge as PW) reached by minimizing a with respect to theany given quantity, that is indistinguishable from lzero, andconsequently is eliminatedfrom the 20 "indicated sum of the squares ofall error factors. i It therefore is evident that by successive mini-Figure 4, while operating fundamentallyin thev am group |25, theaai'group |26, the .am group` |21, the'bi group |28, and the b2 group|29.;are,V

of the :1: variablegroup |2| is the same as group I33Figure 3, exceptthat the output leads |30 and" |3| i d 'l-aff thetaps |38, |36iand |39,and thelead-|3|emay Mllie selectively connected via a five pointselectorf,

|42. For example, using the values shown inthe.. a Voltagefco sin wt,isapplied;to thegrid. The diagram;"the resistor-z|4| lhas'aconduci'iaricemfv outputcurnentthenis.. f

10 micromhos. If this resistor is connected'via 2 2 the switch ma withthe ulead |30, the .5m tap I-I+gme www Sm' wt |38, the O tap |36, the .5tap |39, or the 5 y, y keoz, lead |3|, and considering the potentialofloutput :10+ 'keOLi-'gmeo S111 wt' 99S 210i f conductor |42 as denotedby" e1, the 'currents' ow'e'- ing to conductor |42 Will berespectivelylOx-'lo microamperes, 51e-1061 microamp'eres, microamperes,--5-10e1 microamperes, 10x-1061 microamperes-rl Resstori-^ |48 has aconductance of 2 micromhos. By connecting it viafthe 'selectorl'switch-1 `43 4with the leadsand tapsiltl, tc.,'in the order given#above, currents f or lo frequency butgivesfa large` voltage youtput fortionaletoieolwhich is thefdesiredresul-t. i

of cnductors"'|42 andf '|16 are firstV amplified `by ,t f i, 15 t0` I Aselectively caused to flow through resistor 8 pressed O nv the grids l46 and NM1 of a, pau', of

the common conductor. v| '42?' Since the' above describedl connectionsthrough `f resistorsmi andi my be. ma dem dpendent1y`- platescurrentsof'theilatteritubes are combined'in butfaream parallel, it wm beSeenthaetnefsumeo a Conductofj's'lt owlthrough:a'crcuitreso" of`thefewercurrentsdenverecrfte cnnductor1112Lvl 'nant to 120 Cycles asShow-m0111116 diagram A resistans ten times as greatrespectively-as 25ing tube |54 Whose platecucult 1s also resonant one-tenthitherespectiveconductances. In the '"'f sammanner'as describedaboveg'thes'e-resistoi'sf-l y, l

may* beconne'cted tofyieldI 'combined' currents in 'f spondmg to z 2 bysteps ='off-.1"x milcroamperes; thoughifor 'normaler' to themicro-ammeter |55.

Operation Only"'theIarlgcl'from-r -.9:o1;2i to The .operationofltheadevicefor-solvingvequamainder of thecircuit is a. cathode followerlin- If thi'slcurrent is` applied to anetworkfwliicnf 1061discriminatesi'againstethefD. C.' andthe 'original the second harmonic,Ythe ioutputvvill'be propor# In the present illustrative' circuit the'potentials blocking condenser |5| applies the resultant Jvolt-f f.

`to 120"c"yclesfas shown."l 'From'this'pointi the re*- ear detector"which .leads the ilna'll current .correamperesf.' `By parallel use ofthe three *pairs 'of shown in Figure 3 except, as previouslyfnote'd;.-l

`vthe coeicientryaluesfare enteredldigitallysby set-v ting the selectorswitches inlgroups |24, |25, |26, |21, |28 and |29 manuallygor by anyother suitresistors,l it' is-'evident thatfan'y desired combinedablefmeansalg; currentfivalue --mayf #be obtained N"betweenf'rmicroamperes.v 1 Thus the 'coeicient am 'may -be "'@fdigitally enteredanywherefin the-range 12.99`to 12.995150 thel'se'cond'fractionaldecimal, the 'cure' rent being inf th'eorm a1,1:r:-l3.32e1; ers

The e1 conductor |42 has -branchesl I4 and|75 5"5 Let n be the2numberiof'equations equations balanced to .001;"ffthen=6:;001/11001 50othe presquaring amplification factor Byfthe current law, thesum ofthese, currents must equal zero", andhence as@ the 6'0 cycleamplification of the post-squaring'icircuit'r expressed as a voltage fora given squaringcurrent;u n I v 55 alzo the similar quantity for 120cycles. anequationwhich"determinesthe*voltage^e1z- Then a mustgda'enchcthatL theldierencenin In asimilar manner a-commonf-conductordflphase shifts of the outputs of two distinct preis connected tothevariablefy groups |2-|,"-'|22 and squaring ampliers mustehe less than1 30 for p l23vlajthe fcoeiiicienfgroupsW26;"l21 f anw-129' errors ofless than one partT in a thousand. respectivelyandlfcalrriesa?@potential'correspondi ing-:17 a om e":2 632211!) Y a2,1l=a2,'2yf=bzT='39.96z

Havingthus"'providedvltagsproportional tdw Let V0 be the minimum outputvoltage which e1 and'ez; thepresent'circuit diiers'froxn thatof u ,i5acepi-,ableandmsuppose ..t;he,60 cycle 4output Figure B'inthe-method'oil` securing values ;corre :L nlmst be less than @tenth ofthm Then sponding'f to 12 and'e22. Consider the functional icharacteristicslbt theltubeusuppose .them ,that-.75=Forftlieoperaticnaiksituatiomindicatedmn `the eicients range frornwrl-to 1 andwe wish `our '-f en the minimum voltagesignalrto be squared"circuit, k=.000650 ampere per vvolt squared, g|`n= .000750 ampere pervolt.

Thus

ing are equivalent to variations of nl pre-squaring states, i. e. toaddtwo pcst-squar'ing stages is as effective as adding n pre-squaringstages, one for each equation. However, the circuit discrimination mustalso go up with A.

The values oi the capacities in each resonant circuit are to be adjustedto resonance. There are also RC circuits which can be used, subject tothe limitations stated above. 1

i rents' of tubes |63 and |64, combined in a conductor |16 forming partof acircuit resonated to 120 cycles, have their resultant potentialpassed via a blocking condenser |1| through ya single stage ofamplification also resonant to 120 cycles and including a tube |12, theamplified output being led through a cathode linear follower detector asshown, to a second microammeter |13. It will be seen that thearrangement is essentially the same as that described for the derivationof ei2+ez2,that is, when the voltages .535 and -.5y are impressed on thegrids |65 and |66 of the squaring tubes |63 and |64, the reading of themeter |13 represents a Value proportional to w24-y2.y Obviously the gridinputs could be taken ircmthe r and y output leads themselves instead ofthe .535 and -.5y taps With no basic diiference, the lower input valuesbeing chosen herein for convenience and for keeping the indicatedresultant in proper range on the meter |13.

The principal purpose of the auxiliary circuit just described is tofacilitate the solution ofcer- One must also consider the time delays inthe circuit. These are RC forthe interstage capaci-'1 25 tativecouplingsand for the resonant circuits. `This total time delay shouldnot exceed .25 second. Other-wise the op-` erator is handicapped by theslowness of the response. A

The method of squaring provided in the `circuits shown in Figures. 4 andFlaravoids the plate current In and is also free of drifteiectswhichwould appear in direct current squarelaw detection. 4. i

In the form ofthe invention described above, the coeflicient groups haveemployed resistive conductances. For certain applications the resistiveconductances may be replacedby capaci--y tain types of `problemsinvolving the derivation of the characteristic vectors of a symmetricmatrix. The highly developed mathematical theory involved in thesolution of such problems is well known and need not therefore be setforth herein in detail, .except ,itc state the well known principle thatwith a symmetric matrix all characteristie4 vectors can bederivedprovided either the maximum or minimum vector can' be found.

'Iloiillustrate` the usev of the present invention to locate thelimiting values, take the symmetric i matrix The rows of this matrix areentered in the machine as the coefficient of .r and y in the mannerpreviously described, T being set to zero, thus eliminating the constantb. The output reading tative conductances as generally illustrated in` pFigure 5. In this case, taking thearil group |24a as typical of allcoefficient groups, the leads and taps |36, |3|, |36, |36 and |39 areselectively con45 If the original voltage division remains resistive,that is, determined by resistors |132., |33, etc.,

the wading efeCtS 0f the coefiellt IltWOlk 0n Q0 a maximum, thatis, whenany further variation this division are to the rst order phase shiftsrather than magnitude eie'ctsjand the systemis` less sensitive to theloading. Under" certain cir-i4 cumstances, for example, when higherinput frequencies are to be used, the same eiiect 'as above 6 6 may beobtained by ernployinga'capacitivel volt age division and a resistancenetwork ofthetype shown in Figure 4 in the coefficient groups. Referringagain to Figuresjfl andl 4a', it will be squaring tubes |63 and |164,having their grids |65 and |66 connected valeads |61 and' |68 with the-.5:c output tap |39 ofcvariable group |2| and the similar .511outputtap" |69 ofthey of the meter |55 then represents and bythe abovementioned well-known mathe-` matical theory,` the maximum value of u,under these circumstances corresponds to the values of :c and yrepresenting the maximum characteristic vector of the matrix so long asx2|y2=a same time in such a manner as to keep the above scalereadingconstant, and the.indication of y.

isobserved on the first meter |55, When ,u reaches in .1: and-y kcausesadecrease-in n, `the settings of istie vector or roots of the matrix.

` In a similar manner, varying :I: andy together, with :C24-y2 constant,until p. assumes a minimum on the meter |55, gives there andy valuescorresponding Yto the minimum characteristic vector.

' As previously stated,` whilethe apparatus is illustrated herein asadapted to the solution of seen that the instrument includestwojauxiliary 7,0

variable group |22 respectively.The plate cur-"75 problems of twovariables and equations, it may` f be constructed with provision for anypractical number of variables and coefficients. When constructed formore than two variables the device is" operated in the same manner `as"described to derive the maximum and minimum characteristic' vectors,except that inI the. case of'A more :than twow variab1es,:eachisvaried'with every other Variable inpairsvsuccessively,iwhile.keeping.'the,s-umgof the squaresconstantfluntilthe .fconditionalfmaX-imum or Aminimum :reading-fof p.'is observed onthe meterl.

The structure of the-'device when? equipped 'for anynzdesired number ofvariables and coeilicientsI is generally-the same throughout as shownand describedrherein, the additional required variable and .coeiiicientgroups and their attendant#y con-1 nections,- tubes',.etc.,beingdncorporated-in 'theY samekmannerxto givea reading of n on themeter I55-and-an indicationfofVA the sum of the squares f ofa1l-variables on themeter |13. If desired, an

additional coeiiicientigroup may be incorporated to add'atermA-em to-the:first equation illustrated herein-and aterm--- \y to the secondequation.

this provisionfacilitating the location of other roots between themaximum and minimum characteristics. f.

Whilejthe invention has fbeen described inpreeA ferred-,form `'as noted;*it isv not` limited to the v precses structureiillustratedyias variouschanges*- and. frnodicationsE may :be maderrwithout ldepartk imgTi-roml:the :scope of= the 'appendedclainis What is claimed'fis-f- 1. Thatmethod 4of' solving simultaneous linear.

equations 'having unknownl variables'andknown l' coeicients i associatedtherewith vwhich includes the steps of setting upelectricaluconductances' corresponding :to: said 4coeflicientsz-iriparallel cir-4 eachof said equationslto produce asingleelectri;V

cal value foreach equation proportional to the total error ofsaidequ'ation. due to incorrectness of said'initially entered values'ofsaid variables,

impressing said equation error'values on parallel electroniccircuits;..adapted fto produceoutputs proportional to the squares ofsaid equational errorsg'combining'-said electronic circuit outputs toproduce-fasingle! electrical 'value proportional to the/.sum ofsaidsquaresfv of said'equational f errors, 'measuringsaidlastnamedsingle electri-vcal value to produce-a yquantitative error indication ofsaid sum of said squares, successively varyf ingwthe'settinglo each ofsaid variable'entry values wliilefkeeping the vothers'constant toproduce :final variable' settings 4giving minimum va1 r 2 ues I ot saidftotal errorl indication; and observing saidhiinalisetting or saidvariable entries. f1.-

2. The' method`A claimed. in Claim 1 including the r-step-"ofinitiallyintroducing'insaid combined` circuits' an additionalelectricalentryycomprising-j a gagevariable to bring said'quantitativeerrorl indicationinto" convenient scalar magnitude', and

the "final-step of dividingsaid 4final variable venl ftries bySaid-gagevariable.

3. The method claimed in claiml including`r the steps after completionof'said successive mini;

,65 mizations, of increasing the Velectrical magnitude" of allsaidvariableentries' in a predetermined proportion whereby any. residualerror in said. error indication may be magnified, and refadJusting ,saidvariable settings" successively to minimize said'` magnied errorindication.

4. That method ,Offsolving simultaneous linearV equations whichincludesthe stepslof entering trialelectricalvaluescorresponding totrial values.. of tleiinknowns of. cachot saidequations .into

Inodifying-z-:combination-f witnfoelectri'cal factors corresponding tothe known-:respective coef-efA7V cients of said unknowns, whereby asingle elecA trical value may be established for each of said equationsproportionalrt the'rror therein due to deviation of said trial unknownvalues from the f corre'ctt values# *electronically 1 establishing 'relectrical .'valuesfproportional-rto the squares of l said 'electr-icalAferrorI lvalues, electrically v combin-r 111g saidfisquarederrorv'values' to produce a 'combined'single :errortvalueuproportional to thesumv of thesquares :of said'findividual 1 equation 1 error@I values;andzindividuallyaltering 'said ytrial elec. tricazlfvaluesitoproduce'successive minimizations of saidrsingle combined'er'ror value,"whereby the iina-livalues'lof saidifelectri'calFentriesk producinglsaidr'minimiz'atlonsl-mayf'` proportionally` indicate 1= thecorrectwalues of: saidla'unknowns."

5. In a device for 'solving.simultaneousl'linear equations-havin'gunknown quantity symbols andv knownellcoefficients thereof,"` lincombination,l a

sourcef alternating 7current; a plurality'offtrans` 'l formers connectedto saidffsource-o current and at leastequaliin vnumber to thef-numberrof said comprising an entry means for Evaluesof one of saidunknown quantities, calibrated means to in dividually vary theoutputfvoltages of each of said transformers in propbrtion t0 desirednu- ,iner'icetl entries'ff'of f'f'aid respective unknowns/ meansff'orn'iing separateonductances- Tor'feaclil I of saidcoeicintsandfeaclihailing-af potential'- l output conductor, the conductance means cor-'-spendi-ng unknown'fa'c'tor transformer, lcalibrated variable meansassociated with each of said cone-"-4 ductances fand-f settalilel' in`correspondencel with the@ numerical v'alues' of 4said--respectivecoeicients dto set-'up potentialsfinsaid potential output' conductors'Aproportional Vto said respective unknown factor" entries vmodified-*bysaid respec-l tivev jcoeilicintsffconducting -rneans-v -to combine said'proportional output potentials-to form a single potentialor"each-'ofsaidfequations, electronic means to separately;amplifysaid equationalpof ten-tialsV in yequal'V proportion, electronic v means controllable'by 'said equational potentials'to prodl i ducecurrentsproportional-ttol the-squares oi said equationalpdtentialmeans'toadd rsaid currents, and indicatingmeansito measure said' currentsum.;

yersing switches adapted to selectively interchange the pola-ritiesfbrsad.output conductors of said unknown quantity transformers andreversingswitches. adapted to selectively inter-A change the polarity.connections from said unknown. quantity.`tra1isformerconductors to eachof saidJcoeiilcientconductance means, whereby thesignsof saidA enteredunknown factorsiand coefcintslmaybe selectively,v changed.`

7. devoefas -Clairedwin claim 5 wherein each Vof saidcoefficient'conductancemeans includes'a xedresisto'randthe resistanceme'mber L of aApotentiometer-,of 4matched .total resistance.

connected'.lrespectively` between said twotransformer outputcondctorsand ground, and wherein 'saidicoeicient setting mkeans. includes ,the

movable contact Voflsaid ypoten'tiomet'er connected tosaidpotential'output conductor.

8. -A deviceas'fclair'nedlin claim 5 ,wherein said,v output.conductorslfoi said transformers., area.. bridgedby.agresistorhavingaiplurality of interautres fcmds"apiiiraiityofresistors of graded vresistances, each of said resistors having oneterminal selectively connectible byswitching means to any r one of saidrespective transformer output conductors or taps and its other terminalconnected to said outputpotentialconductor. l;

V9. A device as 4claimed in `claim 5 including means between said sourceof current and said transformers to change the potential delivered bysaid source to,4 said transformers, -wherebylthe indicatedV magnitude ofsaid measuredk Ycurrent sum may be adjusted. .i

10. A device as claimed in claim 5 wherein said output conductors ofeach of said transformers are bridged by a resistor having a pluralityof intermediate taps including a grounded center tap, and wherein eachof said conductance means includes a plurality of condensers ofpredetermined graded capacities, each of said condensers having oneterminal selectively connectable by switching means to any one of saidrespective transformer output conductors or taps and its other terminalconnected to said output potential conductor, said output potentialconductor having a resistive connection to ground.

11. A device as claimed in claim 5 wherein the output `conductors ofeach of said transformers are bridged by a resistor having a pluralityof intermediate taps including a grounded center tap, wherein each ofsaid conductance means in cludes a plurality of resistors ofpredetermined graded resistances, each of said resistors having oneterminal selectively connectable by switching means to any one of saidrespective transformer output conductors and taps and its other terminalconnected to said output potential conductor, and including electronicmeans connected to corresponding output voltage conductors or taps ofsaid transformers and adapted to set up electrical values proportionalto the squares of the potentials of said corresponding transformeroutput conductors or taps, means to add said proportional squaredelectrical values, and second indicating means to measure said lastnamed sum of said squares.

12. That method of solving simultaneous mathematical equations havingunknown variables and known coefcients, which includes the steps ofentering electrical factors proportional to trial unknowns and otherelectrical factors proportional to said known coefficients throughoutsaid equations in an electrical combination to produce a scalarindication proportional to the sum of the squares of the initial errorsin said equations due to incorrectness of said trial electrical entriesand individually changing said trial electrical entries to producesuccessive minimizations of said error indication.

13. In a device for solving simultaneous linear equations having unknownfactors and known coeflicients associated therewith, in combination,electrical means including variable sources of current `and meansforming variable conductances in circuit therewith to establish apotential for each of said equations proportional to the total errorproduced in said equation by entry in said means of electrical factorsrepresentative of said known coefficients and trial values of saidunknown factors, means responsive to said potentials to set upelectrical values proportional to the squares of said errors, means toestablish an elecsquares, and indicating means to measure said value. 4A

` 14. In a device for solving simultaneous linear equations havingunknown factors and known coeiiicients associated therewith, incombination, electrical means including variable sources of current andmeans forming variable conductances connected in circuits therewith toestablish a potential for each of said equations proportional to theerror produced in said equation by entry in said means of electricalfactors representative of said known coeflicients `and trial values ofsaid unknown factors, electronic means responsive to vsaid equationalerror potentials to establish a single electrical value proportional tothe sum of the squares vof said 4equational errors, and indicating meansto measure said single electrical value.

15. A device as claimed in claim 14 wherein said sources of current areof predetermined alternating frequency, wherein said circuits areresonant to said frequency, and wherein said electronic means includesan output circuit resonant to a predetermined harmonic of saidfrequency.

16. A device as claimed in claim 14 wherein said sources of current areof predetermined alternating frequency, wherein said circuits areresonant to said frequency, and wherein said electronic means includesan output circuit resonant to the second harmonic of said predeterminedfrequency.

17. A device as claimed in claim 14 in which said electronic meansincludes diode combinations adapted to provide outputs proportional tothe square of their inputs, said inputs being proportional to saidequational error potentials.

18. A device as claimed in claim 14 in which said electronic meansincludes means to amplify said equational error potentials in equalratio, diode combinations adapted to produce outputs proportional to thesquares of their inputs, said inputs being responsive to said ampliedequational error potentials, and means between said diode combinationsand said indicating means to amplify said single electrical value.

19. That method of deriving the maximum vector of a symmetricmathematical matrix comprising the known coefcients of a set ofsimultaneous linear equations having unknown variable factors associatedwith said coefficients, which includes the steps of entering electricalfactors proportion.. ally representative of said known coeiiicients andtrial values of said unknowns in electrical combination to establish anelectrical value proportional to the algebraic sum of the products ofsaid variable factors and their respective coeiicients for each of saidequations, setting up an electrical value proportional to the sum of thesquares of said algebraic sums, electrically producing an indicationproportional to the magnitude of said sum of said squares, electricallyestablishing a second indication proportional to the sum of the squaresof said trial variable factors unmodied by their respectivecoefficients, and changing said variable factor entries to maximize saidrst indication while maintaining said second indication constant.

20. In a device of the character described, in combination, a source ofcurrent, a plurality of transformers electrically connected to saidsource, means forming individual conductances connected in circuit withthe secondary windings of each of said transformers, adjustable means totap oif a predetermined individual fractional potential from each ofsaid conductances, conducttrical value proportional to the sum of saidins means to combine said. fractional potentials,v

19 ineans forming second 'individual conductanee's connected tosaidvsecondary windings of each of said transformers in respectiveparallel witli said rst conductances, adjustable means to Atap off fromeach of said second conductan'c'es,` conducting means to combine saidsecondfractional-potentials, electronic means responsiveto said combined potentials to establish electrical values proportional to thesquares of said'rst and second combined` potentials, conductive meanstov add said electrical values, means to measure-the'sum of said-addedvalues, and means to individually vary the output voltages o f saidtransformer secondary windings.

FRANCIS J ."MURR-AY.

a predetermined individual fractional potential' 5 20 l REFERENCES;CITE'.. The following references areof recordfiwthe leof this patent:

UNITEDSTAQIES Number Y 'Name Date 2,404,397` Lovell- Jul-y23;-19462,417,098 Wilcox- Man-1,1, 1947 2,446,191 Pemberton Aug. 3, 1948 T-HER:YREFERIENCESl Electronic Computers, by Wma Shannon? an articlein theAugust 1946 issue-lofElec'tronics,pages 110 to 113.

